MATH 613 Modular forms (Topics in Number Theory)

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The course will be largely based on James Milne's notes and F. Diamond and J. Shurman "Introduction to modular forms" (available online at UBC library). We will also somtimes use the classic textbook T. Miyake, "Modular forms" (available online at UBC library). We will occasionally refer to several other sources, including:

• D. Zagier's Lectures on modular forms
• William Stein, "Modular Forms, a computational approach"
• S. Lang, "Introduction to modular forms" and "Elliptic Functions".

Classes: Tue, Th 2-3:30pm in CEME (Civil and Mechanical Engineering), Floor 1, room 1210.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: for now, by appointment.

• Course outline

Announcements

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HOMEWORK

There will be approximately bi-weekly homework assignments.

• Here are some resources if you you need resources for using TeX.
• Problem set 1 (due January 22). Please also read Milne, section on Riemann surfaces as ringed spaces (pp.16-18), including the short section on differential forms.
• Problem set 2 (due February 13). The TeX file of the questions .
• Problem set 3 (Due February 27). The TeX file of the questions

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Final presentations:

The last week of class and possibly a day or two in April will be your lectures. I think it would be reasonable to have 1.5 hours (i.e. a full lecture) per topic, with 2 presenters for each topic, so that you can collaborate (or two 45 min lectures in one class for shorter topics). Please start thinking about what you might like to do. Here are some ideas for topics and some sources (more to be added later):

• The constant term of Eisenstein series and Smith-Minkowski-Siegel mass formula (this relation is also known as the Siegel-Weil formula).
  The last chapter of Serre's "A Course in Arithmetic" (available via UBC library).
• Modular forms of half-integral weight Notes by K. Buzzard
• Elliptic integrals; connection with differential equations. Abel-Jacobi map. Resources:
  • Varadarajan's notes on elliptic functions ;
  • Notes on Abel-Jacobi map
• Algebraic theory of modular forms; possibly the statement of Eichler-Shimura correspondence (see Diamond and Shurman and Chapter II in Milne's notes).
• Any topic that we do not cover in class from Zagier's notes in "The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway" (available online through UBC library).
• Any computational topic (not covered in class) from William Stein's book.
• The monster group and moonshine
• Explicit class-field theory; j-invariant; approximations of pi using modular forms. (resources to be added).

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Detailed Course outline

Short descriptions of each lecture and relevant additional references will be posted here as we progress.

• Tuesday January 6 : Lecture 1. Overview and motivation; the definition of modular forms; Riemann surfaces (approximately matching Milne's Introduction).
• Thursday January. 8 : The goal (to be achieved next class) is to define the structure of a Riemann surface on "H mod Gamma". We did: 1. A quick survey of actions of topological groups (see Milne, Section 1 (pp. 13-14) and Part 1 of the homework 1) with the end result that "H mod Gamma" has Hausdorff quotient topology;
  2. Classification of Fractional-linear transformations, with an aside on realizing the Riemann sphere as the projective line over C (see homework). (Milne, pp.30-31, not including Remark 2.11 which will be discussed later).


• Tuesday January 13 : Lecture 3: 1. Realizing H as a quotient of SL_2(R). (Milne, Proposition 2.1 pp.25-26) 2. the fundamental domain for Gamma(1). (Milne, pp.32-33). 3. The complex structure on H mod Gamma(1). (Milne, pp.35-37, up to but not including the genus computation).
• Thursday Jan. 15 : Lecture 4: Cusps and the complex structure on H^* mod Gamma(1). The complex structure on H^* mod Gamma(N).


• Tuesday Jan. 20 : Lecture 5: Review of the preliminaries on complex analysis on Riemann surfaces. Ringed spaces; sheaves, differential forms. (Milne, pp.16-18). Meromorphic functions on Riemann surfaces (Milne, pp.19-20). Stopped at the definition of divisors and principal divisors. On Thursday will continue from there. If you missed the class, please just read Milne pp.16-21 and you'll be right where we stopped.
• Thursday Jan. 22 : Lecture 6: Review of the analysis/geometry, continued: Riemann-Roch theorem and the notion of genus (Milne, pp.18-23).
  See an illuminating mathoverflow discussion of Riemann-Roch,
  as well as Terry Tao's blog .
  Riemann-Hurwitz formula. (Milne, pp.37-39).


• Tuesday January 27: Lecture 7. Summary of what we did so far; the genus of the modular surface X(N). Started discussing lattices.
• Thursday January 29: Lecture 8. Weierstrass p-function and the structure of an elliptic curve on C/\Lambda. Definition of Eisenstein series. (Milne, pp.43-47).


• Tuesday February 3 : Lecture 9. Lattices, revisited. Equivalence of categories: (lattices mod scaling); (compact Riemann surfaces of genus 1); (elliptic curves over C). Started discussion of modular functions and modular forms; proved Eisenstein series are modular forms and the j-invariant is a modular function. Approximately pp.41-43 and 47-50 in Milne.
  If unfamailiar with discriminants, check out this concise and useful note (or any classic algebra textbook).
• Thursday February 5 : NO CLASS!


• Tuesday February 10: Lecture 10. Modular forms as differentials on H/Gamma. Dimension of the space of modular forms of weight k. Zeroes of modular forms. (Milne, pp.50-54). Also, pay special attention to Proposition 4.3 and Remark 4.4 -- we finished the proof of all the statements in Remark 4.4.
• Thursday February 12: : Lecture 11. Eisenstein series G_2 and G_3 generate the ring of all modular forms. Started Fourier expansion of Eisenstein series. (Approximately pp. 54-57 of Milne). For Fourier expansions of Eisenstein series, see also: Notes by V.S. Varadarajan pp.1-5 for a discussion of Euler's identities. Will finish Fourier expansion of Eisenstein series next class (after the break).


• February 17-19: BREAK.
• Tuesday February 24 - Thursday Feb 26 : Lectures 12-13. The Fourier expansion of the Eisnestein series. The Fourier expansions of the discriminant and the j-invariant. The Jacobi's product formula for the discriminant; Dedekind's eta-function and the Eisenstein series E_1. (See a short section on pp.57-58 of Milne (will also cover Sections 2.3 (skipping the proof of Proposition 6) and 2.4 of Zagier's notes, not including Proposition 8.) An overview of Ramanujan's tau-function and Ramanujan conjecture (will get back to it in some detail later in the course). An overview of some remarkable properties of the j-invariant. There is no specific reference for this part of the lecture but see Milne's Introduction section (p. 11) and the wikipedia article related to the approximations of pi using the j-invariant.
• Tuesday March 3 : Lecture 13. The metric and measure on the upper halpf-plane. Estimates for cusp forms and their coefficients. Petersson inner product. References: Milne, pp. 62-64, or Lang's book on modular forms, chapter 1 and section 2.1; for estimates on cusp forms, see also Zagier's notes, Proposition 8 on p.23.
• Thursday March 5 : Lecture 14. Hecke operators for the full modular group (via lattices). Reference: Milne, pp. 67-79 (we will eventually cover all that's mentioned in these pages, but maybe not quite in the same order). See also Lang's "modular forms" book, chapter 1. One other good source is: Notes on Modular forms by P. Bruin and S. Dahmen
• Tuesday March 10: : Lecture 15. The proof that for the full modular group, the eigenvalues of Hecke operators are algebraic integers (Milne, pp.78-79, Lemma 5.25 to Prop. 5.27, inclusive). L-function attached to a Hecke eingenform and its Euler product; summary of the proof of the multiplicativity of Ramanujan's tau-function.


• Thursday March 12: Lecture 16: Modular forms for congruence subgroups: recap of the dimension formulas for M_k(Gamma) and S_k(Gamma); reminder about cusps; beginnig of the construction of Eisenstein series for Gamma(N). The main reference for this part is Diamond-Shurman, section 4.2 See also Milne, pp.64-66).
  Eisenstein series for Gamma(N); started Hecke operators for Gamma_1(N). Diamond-Shurman, sections 4.3 and 5.1, 5.2
See also Notes on Modular forms by P. Bruin and S. Dahmen , Chapters 4-6.